An introduction to some individual-based methods and to the exercises

Wolf M. Mooij

Netherlands Institute of Ecology

Nt+1 = r Nt (1 - Nt)

The logistic difference equation

Example of a deterministic, non-spatially explicit, unstructured, single species, population dynamical model based on

difference equations

Problem:

Given the research question, how to find the right model from the many models currently

around in ecology

Need for a classification of models

Disclaimer: the one presented here is just one of many possible classifications

How many species do we want to take into account?

• Single species

• Prey predator

• Tritrophic

• Multispecies

• Foodweb

Which part of the life-cycle do we want to take into account?

• Cohort-dynamics (only mortality)

• Population-dynamics (incl. reproduction)

Do we explicitly want to take space into account?

• Spatially implicit

• Spatially explicit -> at what scale?

How do we want to deal with variation between individuals?

• Unstructured model

• Individual-based distribution model (matrix model, physiologically structured model)

• Individual-based configuration model (individual-by-individual model)

Do we want to take stochastic processes into account?

• Deterministic model

• Stochastic model

Which mathematical format are we going to use?

• Difference equations (including matrices) -> at what time scale

• Differential equations (ordinary and partial)

• Logical if-then rules

Nt+1 = r Nt (1 - Nt)

The logistic model implemented as difference equation

• single species• population dynamical• spatially implicit• unstructured• deterministic• based on difference equations

dN/dt = r N (1 – N/K)

The logistic model implemented as differential equation

• single species• population dynamical• spatially implicit• unstructured• deterministic• based on differential equations

The big confusion between differential and difference equations: Euler integration

Differential equation: dN/dt = r Nt (1 – Nt/K)

Euler approximation: Nt+t = Nt + r Nt (1 – Nt/K) t

Difference equation: Nt+1 = Nt + r Nt (1 – Nt/K)

The latter two equations represent different models, especially when rates (r) are high!

dV/dt = r V (1 – N/K) – a V P / (H + V)

dP/dt = a c V P / (H + V) – m P

The Rosenzweig-MacArthur prey-predator model

• prey-predator• population dynamical• spatially implicit• unstructured• deterministic• based on differential equations

Tritrophic, multispecies, and foodweb extensions of these lotka-volterra type of models

• tritrophic, multispecies, foodweb• population dynamical• spatially implicit• unstructured• deterministic• based on differential equations

dF/dt = gmax F (1 – F/K) - N cref W2/3 F2/(F2 + H2)

dN/dt = - m N (1–W/Wmax)p/((1–W/Wmax)p + Lcritp)

dW/dt = e cref W2/3 F2 / (F2 + H2) – rref W2/3

The Scheffer et al. cohort-dynamics prey-predator model

• prey-predator• cohort-dynamics• spatially implicit• unstructured!!!• deterministic• based on differential equations

dF/dt = gmax F (1 – F/K) – N C

dN/dt = - m N Lp/(Lp + Lcritp)

dW/dt = e C - R

The Scheffer et al. cohort-dynamics prey-predator model (version 1.00)

• prey-predator• cohort-dynamics• spatially implicit• unstructured• deterministic• based on differential equations

dF/dt = gmax F (1 – F/K) – N C

Pmort(Ni) = - m Lp/(Lp + Lcritp)

dW/dt = e C - R

The Scheffer et al. cohort-dynamics prey-predator model (version 1.01)

• prey-predator• cohort-dynamics• spatially implicit• unstructured• stochastic• based on differential equations

dF/dt = gmax F (1 – F/K) – Ci

Pmort(Ni) = - m Lip/(Li

p + Lcritp)

dWi/dt = e Ci - Ri

The Scheffer et al. cohort-dynamics prey-predator model (version 1.02)

• prey-predator• cohort-dynamics• spatially implicit• individual-based (configuration with i individuals)• stochastic• based on differential equations

N

dF/dt = gmax F (1 – F/K) – Ci Ni

dNi/dt = - m Ni Lip/(Li

p + Lcritp)

dWi/dt = e Ci - Ri

The Scheffer et al. cohort-dynamics prey-predator model (version 1.03)

• prey-predator• cohort-dynamics• spatially implicit• individual-based (distribution with i classes)• deterministic• based on differential equations

N

The Scheffer et al. cohort-dynamics prey-predator model

deterministic stochastic

unstructured v1.00 v1.01

structured v1.03 v1.02

Advantages and disadvantages of each approach will be discussed during the exercises

Potential of each version

• v1.00 (unstr, determ): Periodic die offs

• v1.01 (unstr, stoch): Extinction probability

• v1.03 (dist, determ): Fitness of strains

• v1.02 (conf, stoch): Rate of strain replacement

Which model is best?

No ind. variation Ind. variation

No Moran effect Synchronized Not synchronized

Moran effect Synchronized Synchronized

Esa Ranta’s Large-scale synchronization model

Nine layers in model construction and analysis

• Language: C++, Visual-Basic

• Libraries: Runga-Kutta integration, etc.

• Frameworks: OSIRIS, EXCEL

• Type: Prey-predator cohort model

• Model: Scheffer et al. model

• Version: Unstr./config./distr., determ./stoch.

• Sensitivity: Parameter H

• Scenario: Brown trout

• Repetition: Demographic stochasticity

OSIRIS: Object-oriented Simulation Framework for Individual-based Simulations

“A personal solution to a shared problem”

Developed and used as a personal research tool, now also used in cooperation with others

Written in C++ (ca. 10000 lines)

Main aspects: predefined simulation objects, event queue, standardized input and output

Interface via EXCEL (through ASCII files)

Five predefined simulation objects

• Analyser (as many as there are analysis)

• System (one only)

• Habitat (as many as there are spatial units)

• Population (as many as there are species)

• Individual (as many as there are individuals)

Analyser

System

Habitat Population

Individual

datatasks

datatasks

datatasks

datatasks

datatasks

Simulating dynamics with an event queue

state at time t dynamics state at time t+t

task id 1 perform task 1 task id 3

task id 2 task id 2

task id 3 task id 1

task id 4 task id 4

task id 5 task id 5

OSIRIS library

PCLake code

Excel

Interface code

Borland C++

Input workbook

Notepad

Tools library

Log workbook

Output workbook

Report workbook

Input files

Output files

Executable